Posted by: Alexandre Borovik | May 26, 2008

Calculus without limits

I re-publish my old post from my old blog, with comments (some of them new) included in the body.

My post:

Thanks to Michael Livshits, I became aware of an alternative approach to calculus based on eliminating the concept of a limit and replacing it by uniform Lipschitz bounds. For example, definition of derivative becomes

|f(x)-f(a)-f'(a)(x-a)| < K(x-a)^2

I would love to learn more. Any comments?

“Calculus without limits” T-shirt is available from Michael’s e-shop, take a look.

Comments from the old blog:

Torus said…
The book of Marsden and Weinstein “Calculus Unlimited” is available here. It has a similar approach.
Anonymous said…
Dear Alexandre, I appreciate your interest in my efforts to simplify Calculus. In addition to many links from my home page at, there is a book by a Chinese mathematician Qun Lin called “Free Calculus: A Liberation from Concepts and Proofs,” ISBN 9789812704597, published in 2007 by World Scientific, which is based on similar ideas, summarized in his paper at
There are also some articles by Mark Bridger at
as well as “The Lost Calculus: Tangency and Optimization without Limits” by Jeff Suzuki, Mathematics Magazine, December 2005, and very recent one by Harold Edwards, dedicated to Euler’s tricentenial, at

In reply to torus: while “Calculus Unlimited” is a fine attempt to introduce differentiation without using limits, it is still based on differentiability at a given point, and needs some heavy tools like completeness and compactness to get any practical results.

For everybody else: the correct formula in Alexandre’s original post is |f(x)-f(a)-f'(a)(x-a)| is less or equal to K|x-a|^2

I am Миша Лившиц and I approve this message. The views expressed are my own and not of any organization (I hate them all).

Anonymous said…
“Free Calculus: A Liberation from Concepts and Proofs”

Why would anyone want to be “liberated” from proofs? Surely, mathematical proof is one of the supreme and most beautiful achievements of human culture!

Anonymous said…
Not all of them, silly, only the superfluous ones, the ones that just clog up your brain.

Zen Harper said…
This definition is definitely incorrect; it does not work for all differentiable functions f.

For a simple example, let f(x)=|x|^p with p strictly between 1 and 2, and a=0=f(a)=f'(a). Then f’ exists (and is continuous) on the whole real line, but there is no fixed K such that |f(x)| is less than Kx^2 for all sufficiently small x.

Everyone has their own opinion on calculus teaching, but I personally feel that the classical methods and proofs of standard Analysis are already about as close to perfection as undergraduate maths gets, in terms of beauty and simplicity. If students can’t get it, or aren’t interested, they should switch to Physics or Engineering…

(Of course, you may have the perfectly valid opinion that Analysis and Calculus are different subjects; but I disagree, at least as far as mathematics students are concerned. University is the time to start learning real maths properly, and to undo the terrible damage caused by GCSE and A-level, i.e. tedious memorisation of methods without proof or understanding).

Alexandre Borovik said…
Zen Harper: it is a not wrong definition, it is an alternative definition. Yes, it sacrifices functions like the one sugested by you. But it allows to develop a consistent theory with full proofs, even if concerned with narrower classes of funcions.
misha said…
Zen Harper: Take your favorite modulus of continuity m, then replace K(x-a)^2 in the definition by Km(|x-a|)(|x-a|). Enjoy the simplicity and directness of all the proofs. See more discussion on the next reincarnation of this blog.

You also said: “… I personally feel that the classical methods and proofs of standard Analysis are already about as close to perfection as undergraduate maths gets, in terms of beauty and simplicity. If students can’t get it, or aren’t interested, they should switch to Physics or Engineering…”

This is the most arrogant piece of nonsense that I’ve seen in a while. First of all, Calculus and Analysis are not carved in stone, there is a lot of room for improvement, both conceptual and pedagogical. Just compare the proofs in both approaches and see the difference. Second, most of the students who are forced to take Calculus, will use it as a tool for applications, and they are not interested in mathematical hair-splitting, they want to understand the practical aspects of the subject. It is the indiscriminate shoving of the mostly irrelevant mathematical formalism down the students’ throats that breeds the “tedious memorisation of methods without proof or understanding,” as you have aptly put it. And finally, the approach based on explicit uniform estimates, that I and other people suggest, brings Calculus into the realm of modern mathematics, puts Classical Analysis in its proper context and makes it easier to understand even for the math majors. I rest my case.

Zen Harper said…
I was surprised to see such a strong backlash against my comments. I am not trying to start a war here, I am just stating my opinion. Maybe we will just have to agree to disagree on this one.

I’m sorry if you think I sound arrogant. Maybe it came across slightly wrong because I am thinking in terms of the English university system, where students don’t have “majors” and “minors” – they make a choice (before university) whether to study a degree in mathematics, physics, engineering, or something else.

My point is that Mathematics students (in the English system I am talking about) are there to learn Mathematics, and that means proper, undiluted rigorous proofs in great generality; the “pathological” differentiable functions, for which your proposed alternatives fail, are admittedly not needed for most purposes in physics, engineering, and other applications. But they are useful sometimes (even in applications! e.g. random walks, Brownian motion, probability, functional analysis, etc. etc. etc.) and mathematicians definitely do need them; the longer students delay in seeing them, the harder it will be for them to get used to the Analytic way of thinking, and understanding of “pathological” functions.

If instead you’re talking about Calculus courses for non-mathematics students in the American system, then that’s an entirely different set of students, and not the one I was talking about. I suppose I should have made this clear.

Another very important point, which I forgot to say precisely because it normally goes without saying, is that “differentiable function” is a totally standard phrase, with a standard meaning, agreed upon by over 99% of all mathematicians. You really shouldn’t use standard terminology to refer to something else; this will only confuse students more when they read other books (and is totally contrary to accepted practice).

It’s fine to use alternative approaches, but then you must use alternative terminology also; you should tell students the classical definitions (even if you are not planning to use them); and you should point out the negative aspects of your alternative approach.

Otherwise, you are simply being dishonest to your students. You have a duty to teach students mathematics; if you want to use your own non-standard variants, you must make it clear to them.

In reply to Misha, about choosing a modulus of continuity m: I agree completely that it simplifies some proofs. But it still doesn’t give you all differentiable functions! As soon as you make a choice about which m to use, and fix it, you are then restricting to a subclass of differentiable functions. As you make m tend to zero more slowly, you make the class of functions larger; but you will never be able to deal with all differentiable functions SIMULTANEOUSLY, whatever fixed choice of m you make. If instead you allow m to change depending on the function f, which has already been in common use in standard textbooks for decades, then (as is well-known and standard) you get simply a trivial rephrasing of the classical definition.

In summary: really, the only thing I object to is the use of the word “differentiable”. Just make up your own alternative word instead, and no-one will complain.

Zen Harper said…
P.S. to my previous comment, with a stupid mistake at the end:

Of course, the modulus m must depend not just on f, but also on the point a, in order to recover the full class of differentiable functions.

misha said…
To Zen Harper: Brownian motion trajectories are (locally) uniformly Holder continuous with any exponent less than 1/2. When you get to functional analysis, you can use distributions and flush your classical differentiation theory down the toilet, it’s a mess anyway.Talking about terminology, I call the functions differentiable in the stronger sense uniformly Lipschitz differentiable, or uniformly Holder differentiable, accorrding to the modulus of continuity, so nobody is misled. Since any continuous function on a closed finite interval has a uniform modulus of continuity, we can capture any continuously differentiable function with this approach, and these are the most important. Peter Lax used uniform differentiability in his calculus book of 1976, and Mark Bridger has recently published a text book on introductory analysis, based on uniform notions.

And who needs to deal with ALL the diferentiable functions SIMULTANEOUSLY? Can you give an example? Similarly, in most analytic applications, you need more than continuity to get any meaningful results, “general” continuous functions are highly pathological.

Finally, I don’t suggest to totally throw away the classical definitions of pointwise continuity and pointwise differentiability (althought they are mostly inadequate and need compactness and other crutches to get to anything useful). I’m just saying that starting with them in calculus and introductory analysis is not very clever, especially in calculus, where these lofty notions are almost never used and never properly explained, and you just end up looking as a fool, paying a lip service to these notions and giving complicated or vague explanations, that only confuse your students, while much simpler and clear explanations are available.

Look, when you study or teach a new subject, it’s a bad idea to start with the most general definitions and pathologies, you start with the simple stuff that is immediately useful in solving problems and applicable to the other subjects, you can get to generalities and pathologies later, and they will make more sense after you or your students get a good grasp of simple examples and applications.

Zen Harper said…
To Misha again…

On this particular subject, I sympathise with the saying “if it ain’t broke, don’t fix it…” Really, what’s wrong with the traditional limits approach?

Just a quick example: the Lebesgue Differentiation Theorem, as far as I know, gives almost no information beyond mere pointwise differentiability (only almost everywhere). I only mentioned Brownian motion as an example of “natural” but non-differentiable continuous functions; I wasn’t thinking in terms of additional continuity properties.

It seems to me that the question “what is the best way to teach calculus/analysis?” is both subjective and highly dependent upon the target set of students. We still disagree on a number of issues; I cannot see why you seem to be so reluctant to use limits in the first course, and why you think the modulus of continuity approach is so much easier for students. I think we both agree, however, that understanding of limits is essential, and definitely needs to be done at some point; so when, exactly, do you propose to teach it? The second year? The third year? That doesn’t leave very much time for more advanced courses.

Also, I would say (although this is quite subjective) that “continuity” is more “mathematically natural” than continuity plus Holder or Lipschitz estimates; and so, similarly, pointwise differentiability is more “natural” than differentiability plus additional estimates. Your approach emphasises “unnatural” estimates, which are not “typical” of “most” functions.

There is a strong danger that students will mistake your special Holder and Lipschitz estimates (which are still, of course, very useful for some purposes) for being “typical”.

I don’t agree at all with your final paragraph (although, of course, it is a matter of taste). I think students will not appreciate your approach, in the long run, precisely BECAUSE you delay too long in showing them the nasty examples! A similar thing happens with the famous exp(-1/x^2) function, which confuses most students who have been so strongly conditioned to think in terms of Taylor series. In the long term, I think it is better for students to see such things as soon as possible. Just try doing a quick survey of third year undergraduates in any good UK university, to find out how many of them are aware of such nasties (even without knowing the detailed examples); I bet you will find a surprisingly large number who aren’t.

Of course, you don’t necessarily have to spend a large amount of time on the “pathologies” (many of which, like exp(-1/x^2), are not really pathological at all); mere knowledge of their existence is already enough for many purposes.

Here are some of my axioms:

AXIOM 1: Mathematics is very hard, and unavoidably needs serious work.

I think most of us would agree with this, if we are being honest.

AXIOM 2: The main purpose of a Mathematics B.A. or B.Sc. (or M.Math.) degree SHOULD be to teach students enough mathematics to prepare them for further study or research (either in mathematics, or in a highly mathematical “applied” field).

It is NOT to train future computer programmers, bankers, accountants, airline pilots, etc. etc. how to do their jobs (for which a mathematics degree is almost a total waste of time, if we are being honest). Such people only study mathematics for fun; and it is DEFINITELY not to provide basic numeracy skills (for which GCSE or A-level maths is already more than adequate). University lecturers (I am one myself) are about the worst possible people to give vocational training, and we should stop trying to pretend otherwise.

AXIOM 3: a university degree, whatever the subject, SHOULD be very difficult and demanding. A secondary purpose is to “toughen up” students, similarly to hard physical training for athletes. Another purpose is to actually show students what MATHEMATICS is; unfortunately, GCSE and A-level are totally watered down and inadequate nowadays, and do not really contain any serious maths at all.

AXIOM 4: three or four years is, in fact, only a very short amount of time; far less than the optimal pre-Ph.D. time. There is no time to be wasted!

…anyway, now that I’ve stated my axioms I can defend my position some more: first, limits undeniably must be covered (Axiom 2). You can’t avoid them for too long (Axiom 4), so they really should be taught no later than the first undergraduate year. The fact that students find limits difficult is completely irrelevant (by Axioms 1 and 3). By Axiom 4, it would not be completely unreasonable if you only had enough time for one approach: EITHER the standard limits approach, OR your modulus of continuity approach. Which should be omitted? (Of course the optimal course is to cover both; but then, the world we live in is far from optimal).

I still think (coming back to Axioms 1,2,3) that any student who understands limits properly will then have no difficulty understanding your alternative approach; but any student who can’t understand limits will also not understand your approach (they themselves might THINK they do, but REALLY they don’t; how many students REALLY understand uniform continuity or term-by-term differentiation of power series, for example?)


  1. To Zen Harper:

    If you think it’s not broke, especially outside of the pure math department, you are in denial. Everybody complains about it, it’s an elephant in the room that you choose to ignore.

    As for Lebesque differentiation theorem (that says that any monotonic function is differentiable almost evrywhere), that certainly does not belong to the first semester analysis course, so I’m not sure why you are bringing it up, it’s showcase example of how classical differentiation doesn’t quite work with Lebesque integral, because if you integrate the derivative, you will recover only the absolutely continuous part of your monotonic function. To reconstruct any pointwise differentiable function from its derivative you either need the Henstock-Kurzweil integral, or you have to look at your function as a distribution. The classical (19th century) analysis works well only with (piecewise) continuously differentiable bunctions, i.e., piecewise uniformly differentiable functions.

    Limits of functions and pointwise continuity can be treated after the theory based on uniform estimates is developed, and it will not take less time than in the traditional approach since the ground is well prepared. So the time you spend on the streamlined theory is not wasted, as you may argue, but serves as a good preparation for the more advanced topics. See an essay by Hermann Karcher for more details. By the way, he said that the students who learned it this way, reported less problems with more advanced topics, such as numerical analysis.

    Why do you say “continuity plus Holder or Lipschitz estimates?” It’s like saying “continuity plus differentiability,” I would not expect it from a mathematician, but it’s a minor point. As for being “typical,” it depends on typicality. In my opinion, non-Holder continuous functions are on the fringe of mainstream mathematics. There is no danger at all that students will think that all continuous functions are Holder or Lipschitz, especially if you point it out and give a few examples.

    Now about your AXIOMS and the implications that you derive.

    AXIOM 1. Yes, mathematics is hard but it doesn’t mean you have to make it even harder by starting with the hard topics. And it doesnt mean that making the ideas of calculus more accessible is not worth the effort. By articially making mathematics less accessible than it has to be, you really shoot yourself in the foot, as a mathematician.

    AXIOM 2 Doesn’t mean the ideas of calculus should be taught only in the orthodox 19th century way that you advocate. And be not taught at all to non-specialists.

    AXIOM 3 Doesn’t mean that the university degree should be made even harder by rather thoughtless and rigid arrangement of the material in undergraduate analysis.

    AXIOM 4 Doesn’t mean that starting with uniform estimates is a waste of time, most likely, it will save some time, even during the first year, and lead to better results at that.

    Your position seems to be that everything is hunky-dory the way it is, and any objections are irrelevant. I’m not buying it. Your claim that the students who have difficulties with the canonical approach will have no less difficulties with a simplified approach doesn’t hold water, axioms or no axioms.

  2. In the third paragraph of my previous post “…and it will not take less time…” should be “…and it will take less time…,” sorry for sloppy editing. I’ll post later on yet another way to look at differentiation, as division of f(x)-f(a) by x-a in a certain ring, the way that may appeal to math majors.

  3. Here we go. Nobody needs limits to differentiate polynomials, since we can explicitly divide p(x)-p(a) by x-a, as polynomials in x, and stick x=a into the result of the division to get p'(a). Similarly, the Cauchy’s definition of differentiability of f at a is divisibility of f(x)-f(a) by x-a in the ring of functions of x continuous at a (a is kept constant).

    Now, we can work with the other rings (and modules) to get the other flavors of differentiability. If we view p(x)-p(a) and x-a as functions of 2 variables and our ring is the ring of unifomly continuous functions of 2 variables, we get the uniform differentiability. Taking Lipschittz functions of 2 variables will gives us the uniform Lipschitz differentiability with the inequality |f(x)-f(a)-f'(a)(x-a)|\leq K(x-a)^2 that was the starting point of this discussion. Instead of Lipschitz, we can use Holder, or the class related to some other modulus of continuity. It even works for distributions.

    This view of differentiation gives us a broader perspective, shows us how to simplify calculus, and also extends to many variables, complex variables and even more general situations. Some people say it is a crime against Analysis, but I don’t see it this way.

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